A physics student wishes to determine the density of an unknown metal cylinder.

Question

A physics student wishes to determine the density of an unknown metal cylinder. She makes the following measurements for the cylinder's mass, length and diameter: (a) Which of theses three measurements is the most precise? M L D cannot be determined. (b) Determine the density of the cylinder, the uncertainty in that density (using the rules of error propagation) and the relative (percent) uncertainty of the density. Density = kg/m3 Uncertainty in the Density = kg/m3 Relative (Percent) Uncertainty in the Density = % What is the relative (percent) uncertainty of each of the three original measurements? dM/M x 100 = % dL/L x 100 =% dD/D x 100 = %

Explanation / Answer

a.) most precise is M since dM = 0.38 which is greater than dL = 0.1 and dD = 0.3

b.) d = M/V

V = (pi/4)*D^2*L

So d = (4/pi)*(M/D^2L)

Density, d = (4/pi)*(9.61x10^-3/(0.023^2*0.054))

So Density, d = 428.34 Kg/m^3

(d(d)/d) = 2*(dD/D) + (dL/L)

d(d)/d = 2*(0.3/2.3) + (0.1/5.4) = 0.2794

So d(d) = 428.34*0.2794 = 119.67 Kg/m^3

Uncertainity in density = 119.67 Kg/m^3

Relative Uncertainity = d(d)/d = 0.2794 = 27.94%

c.)

dM/M = 0.38/9.61 = 0.0395 = 3.95%

dL/L - 0.1/5.4 = 0.0185 = 1.85%

dD/D = 0.3/2.3 = 0.13 = 13%

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